Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64)
gp: K = bnfinit(y^18 + 6*y^16 - 24*y^15 - 72*y^14 + 132*y^13 + 774*y^12 + 828*y^11 - 102*y^10 - 416*y^9 - 324*y^8 - 1752*y^7 - 1515*y^6 + 1476*y^5 + 1632*y^4 - 576*y^3 - 288*y^2 + 192*y + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64)
\( x^{18} + 6 x^{16} - 24 x^{15} - 72 x^{14} + 132 x^{13} + 774 x^{12} + 828 x^{11} - 102 x^{10} + \cdots + 64 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-781553895537094671760883712\)
\(\medspace = -\,2^{18}\cdot 3^{31}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.19\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3/2}3^{97/54}13^{2/3}\approx 112.51878599744718$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{3}+\frac{1}{3}$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{4}+\frac{1}{3}a$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{12}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{24}a^{13}-\frac{1}{24}a^{12}+\frac{1}{24}a^{10}+\frac{1}{24}a^{9}-\frac{1}{8}a^{7}+\frac{3}{8}a^{6}+\frac{1}{4}a^{5}+\frac{5}{24}a^{4}-\frac{5}{24}a^{3}-\frac{1}{4}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{72}a^{14}+\frac{1}{72}a^{13}-\frac{1}{36}a^{12}+\frac{5}{72}a^{11}-\frac{1}{72}a^{10}+\frac{1}{36}a^{9}-\frac{5}{24}a^{8}+\frac{1}{24}a^{7}-\frac{1}{3}a^{6}-\frac{19}{72}a^{5}+\frac{17}{72}a^{4}-\frac{2}{9}a^{3}+\frac{7}{18}a^{2}+\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{72}a^{15}-\frac{1}{36}a^{12}-\frac{1}{12}a^{10}+\frac{1}{18}a^{9}-\frac{11}{36}a^{6}-\frac{1}{4}a^{4}-\frac{1}{72}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a-\frac{2}{9}$, $\frac{1}{1008}a^{16}-\frac{1}{252}a^{15}+\frac{1}{252}a^{14}-\frac{5}{504}a^{13}+\frac{1}{28}a^{12}-\frac{5}{72}a^{11}+\frac{1}{63}a^{9}-\frac{2}{21}a^{8}+\frac{85}{504}a^{7}+\frac{97}{252}a^{6}-\frac{137}{504}a^{5}-\frac{389}{1008}a^{4}-\frac{5}{21}a^{3}+\frac{16}{63}a^{2}+\frac{10}{21}a+\frac{8}{63}$, $\frac{1}{54\!\cdots\!56}a^{17}+\frac{177614821203493}{91\!\cdots\!76}a^{16}+\frac{16\!\cdots\!29}{30\!\cdots\!92}a^{15}+\frac{888802893648087}{15\!\cdots\!96}a^{14}-\frac{602941283376823}{34\!\cdots\!16}a^{13}-\frac{31\!\cdots\!41}{13\!\cdots\!64}a^{12}+\frac{20\!\cdots\!69}{39\!\cdots\!04}a^{11}+\frac{31\!\cdots\!61}{17\!\cdots\!58}a^{10}-\frac{13\!\cdots\!75}{27\!\cdots\!28}a^{9}-\frac{88\!\cdots\!95}{13\!\cdots\!64}a^{8}+\frac{19\!\cdots\!79}{21\!\cdots\!28}a^{7}+\frac{43\!\cdots\!93}{22\!\cdots\!44}a^{6}+\frac{12\!\cdots\!67}{18\!\cdots\!52}a^{5}+\frac{68\!\cdots\!81}{27\!\cdots\!28}a^{4}+\frac{23\!\cdots\!77}{19\!\cdots\!52}a^{3}-\frac{13\!\cdots\!51}{68\!\cdots\!32}a^{2}-\frac{12\!\cdots\!65}{34\!\cdots\!16}a-\frac{17\!\cdots\!41}{17\!\cdots\!58}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{4801125193}{114590730048} a^{17} - \frac{170632683}{19098455008} a^{16} + \frac{15036152261}{57295365024} a^{15} - \frac{10078936511}{9549227504} a^{14} - \frac{39110310367}{14323841256} a^{13} + \frac{170104184185}{28647682512} a^{12} + \frac{1741847757259}{57295365024} a^{11} + \frac{413765020993}{14323841256} a^{10} - \frac{60908827561}{19098455008} a^{9} - \frac{122394471931}{28647682512} a^{8} - \frac{29463888015}{9549227504} a^{7} - \frac{471986451481}{7161920628} a^{6} - \frac{1726290497513}{38196910016} a^{5} + \frac{3423014335129}{57295365024} a^{4} + \frac{1013633713667}{28647682512} a^{3} - \frac{506218354741}{14323841256} a^{2} + \frac{5221075213}{7161920628} a + \frac{7512133399}{1193653438} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{66\!\cdots\!53}{13\!\cdots\!68}a^{17}-\frac{5742289198511}{11\!\cdots\!14}a^{16}+\frac{61\!\cdots\!93}{19\!\cdots\!52}a^{15}-\frac{50\!\cdots\!65}{40\!\cdots\!49}a^{14}-\frac{56\!\cdots\!29}{16\!\cdots\!96}a^{13}+\frac{66\!\cdots\!03}{98\!\cdots\!76}a^{12}+\frac{35\!\cdots\!81}{93\!\cdots\!12}a^{11}+\frac{42\!\cdots\!21}{10\!\cdots\!64}a^{10}-\frac{24\!\cdots\!57}{19\!\cdots\!52}a^{9}-\frac{54\!\cdots\!33}{81\!\cdots\!98}a^{8}-\frac{20\!\cdots\!41}{32\!\cdots\!92}a^{7}-\frac{40\!\cdots\!11}{49\!\cdots\!88}a^{6}-\frac{85\!\cdots\!03}{13\!\cdots\!68}a^{5}+\frac{21\!\cdots\!05}{32\!\cdots\!92}a^{4}+\frac{12\!\cdots\!05}{24\!\cdots\!94}a^{3}-\frac{55\!\cdots\!33}{16\!\cdots\!96}a^{2}-\frac{283548111819103}{27\!\cdots\!66}a+\frac{76\!\cdots\!26}{12\!\cdots\!47}$, $\frac{16\!\cdots\!85}{18\!\cdots\!52}a^{17}-\frac{11\!\cdots\!61}{91\!\cdots\!76}a^{16}+\frac{15\!\cdots\!51}{27\!\cdots\!28}a^{15}-\frac{10\!\cdots\!17}{45\!\cdots\!88}a^{14}-\frac{14\!\cdots\!19}{22\!\cdots\!44}a^{13}+\frac{17\!\cdots\!07}{13\!\cdots\!64}a^{12}+\frac{88\!\cdots\!13}{13\!\cdots\!68}a^{11}+\frac{51\!\cdots\!29}{76\!\cdots\!48}a^{10}-\frac{14\!\cdots\!13}{27\!\cdots\!28}a^{9}-\frac{64\!\cdots\!59}{45\!\cdots\!88}a^{8}-\frac{64\!\cdots\!11}{65\!\cdots\!84}a^{7}-\frac{50\!\cdots\!45}{34\!\cdots\!16}a^{6}-\frac{20\!\cdots\!99}{18\!\cdots\!52}a^{5}+\frac{11\!\cdots\!49}{91\!\cdots\!76}a^{4}+\frac{18\!\cdots\!65}{19\!\cdots\!52}a^{3}-\frac{16\!\cdots\!27}{22\!\cdots\!44}a^{2}-\frac{12\!\cdots\!93}{38\!\cdots\!24}a+\frac{25\!\cdots\!23}{17\!\cdots\!58}$, $\frac{19\!\cdots\!11}{13\!\cdots\!64}a^{17}+\frac{257207468520799}{22\!\cdots\!44}a^{16}+\frac{60\!\cdots\!49}{68\!\cdots\!32}a^{15}-\frac{28\!\cdots\!38}{85\!\cdots\!29}a^{14}-\frac{73\!\cdots\!95}{76\!\cdots\!48}a^{13}+\frac{11\!\cdots\!81}{68\!\cdots\!32}a^{12}+\frac{10\!\cdots\!47}{98\!\cdots\!76}a^{11}+\frac{27\!\cdots\!83}{22\!\cdots\!44}a^{10}+\frac{24\!\cdots\!24}{85\!\cdots\!29}a^{9}+\frac{38\!\cdots\!51}{34\!\cdots\!16}a^{8}+\frac{18\!\cdots\!51}{32\!\cdots\!92}a^{7}-\frac{14\!\cdots\!99}{68\!\cdots\!32}a^{6}-\frac{26\!\cdots\!89}{13\!\cdots\!64}a^{5}+\frac{26\!\cdots\!57}{19\!\cdots\!62}a^{4}+\frac{11\!\cdots\!65}{98\!\cdots\!76}a^{3}-\frac{79\!\cdots\!10}{85\!\cdots\!29}a^{2}-\frac{26\!\cdots\!25}{57\!\cdots\!86}a+\frac{11\!\cdots\!45}{85\!\cdots\!29}$, $\frac{42\!\cdots\!63}{91\!\cdots\!76}a^{17}-\frac{770012263621417}{11\!\cdots\!72}a^{16}+\frac{35\!\cdots\!61}{13\!\cdots\!64}a^{15}-\frac{35\!\cdots\!97}{22\!\cdots\!44}a^{14}-\frac{53\!\cdots\!41}{28\!\cdots\!43}a^{13}+\frac{95\!\cdots\!04}{85\!\cdots\!29}a^{12}+\frac{19\!\cdots\!01}{65\!\cdots\!84}a^{11}-\frac{29\!\cdots\!49}{22\!\cdots\!44}a^{10}-\frac{10\!\cdots\!61}{13\!\cdots\!64}a^{9}-\frac{14\!\cdots\!89}{22\!\cdots\!44}a^{8}-\frac{14\!\cdots\!29}{32\!\cdots\!92}a^{7}-\frac{68\!\cdots\!11}{68\!\cdots\!32}a^{6}+\frac{25\!\cdots\!47}{91\!\cdots\!76}a^{5}+\frac{47\!\cdots\!49}{22\!\cdots\!44}a^{4}+\frac{17\!\cdots\!49}{24\!\cdots\!94}a^{3}-\frac{23\!\cdots\!33}{28\!\cdots\!43}a^{2}+\frac{44\!\cdots\!59}{28\!\cdots\!43}a+\frac{16\!\cdots\!42}{85\!\cdots\!29}$, $\frac{60\!\cdots\!69}{54\!\cdots\!56}a^{17}+\frac{16\!\cdots\!39}{27\!\cdots\!28}a^{16}+\frac{29\!\cdots\!25}{27\!\cdots\!28}a^{15}+\frac{20\!\cdots\!81}{13\!\cdots\!64}a^{14}-\frac{13\!\cdots\!03}{68\!\cdots\!32}a^{13}-\frac{50\!\cdots\!49}{13\!\cdots\!64}a^{12}+\frac{50\!\cdots\!93}{39\!\cdots\!04}a^{11}+\frac{39\!\cdots\!49}{68\!\cdots\!32}a^{10}+\frac{23\!\cdots\!69}{27\!\cdots\!28}a^{9}+\frac{85\!\cdots\!91}{13\!\cdots\!64}a^{8}+\frac{80\!\cdots\!77}{19\!\cdots\!52}a^{7}+\frac{61\!\cdots\!59}{68\!\cdots\!32}a^{6}-\frac{51\!\cdots\!51}{54\!\cdots\!56}a^{5}-\frac{33\!\cdots\!03}{27\!\cdots\!28}a^{4}-\frac{18\!\cdots\!49}{19\!\cdots\!52}a^{3}+\frac{23\!\cdots\!75}{68\!\cdots\!32}a^{2}-\frac{53\!\cdots\!43}{34\!\cdots\!16}a-\frac{17\!\cdots\!29}{17\!\cdots\!58}$, $\frac{23\!\cdots\!45}{49\!\cdots\!88}a^{17}-\frac{19\!\cdots\!83}{98\!\cdots\!76}a^{16}+\frac{14\!\cdots\!99}{49\!\cdots\!88}a^{15}-\frac{62\!\cdots\!63}{49\!\cdots\!88}a^{14}-\frac{36\!\cdots\!24}{12\!\cdots\!47}a^{13}+\frac{18\!\cdots\!85}{24\!\cdots\!94}a^{12}+\frac{60\!\cdots\!55}{17\!\cdots\!21}a^{11}+\frac{12\!\cdots\!29}{49\!\cdots\!88}a^{10}-\frac{37\!\cdots\!75}{24\!\cdots\!94}a^{9}-\frac{68\!\cdots\!21}{49\!\cdots\!88}a^{8}-\frac{17\!\cdots\!23}{17\!\cdots\!21}a^{7}-\frac{98\!\cdots\!75}{12\!\cdots\!47}a^{6}-\frac{19\!\cdots\!23}{49\!\cdots\!88}a^{5}+\frac{86\!\cdots\!85}{98\!\cdots\!76}a^{4}+\frac{30\!\cdots\!09}{70\!\cdots\!84}a^{3}-\frac{55\!\cdots\!49}{12\!\cdots\!47}a^{2}+\frac{52\!\cdots\!48}{12\!\cdots\!47}a+\frac{92\!\cdots\!19}{12\!\cdots\!47}$, $\frac{23\!\cdots\!57}{27\!\cdots\!28}a^{17}-\frac{57\!\cdots\!07}{17\!\cdots\!58}a^{16}+\frac{79\!\cdots\!81}{15\!\cdots\!96}a^{15}-\frac{76\!\cdots\!81}{34\!\cdots\!16}a^{14}-\frac{89\!\cdots\!11}{17\!\cdots\!58}a^{13}+\frac{10\!\cdots\!85}{76\!\cdots\!48}a^{12}+\frac{11\!\cdots\!77}{19\!\cdots\!52}a^{11}+\frac{31\!\cdots\!83}{68\!\cdots\!32}a^{10}-\frac{12\!\cdots\!73}{45\!\cdots\!88}a^{9}-\frac{20\!\cdots\!96}{85\!\cdots\!29}a^{8}-\frac{17\!\cdots\!65}{98\!\cdots\!76}a^{7}-\frac{54\!\cdots\!59}{38\!\cdots\!24}a^{6}-\frac{19\!\cdots\!67}{27\!\cdots\!28}a^{5}+\frac{10\!\cdots\!37}{68\!\cdots\!32}a^{4}+\frac{11\!\cdots\!41}{15\!\cdots\!52}a^{3}-\frac{13\!\cdots\!29}{17\!\cdots\!58}a^{2}+\frac{62\!\cdots\!08}{85\!\cdots\!29}a+\frac{38\!\cdots\!11}{28\!\cdots\!43}$, $\frac{83\!\cdots\!75}{17\!\cdots\!58}a^{17}+\frac{19\!\cdots\!77}{11\!\cdots\!72}a^{16}+\frac{18\!\cdots\!05}{68\!\cdots\!32}a^{15}-\frac{73\!\cdots\!61}{68\!\cdots\!32}a^{14}-\frac{45\!\cdots\!19}{11\!\cdots\!72}a^{13}+\frac{39\!\cdots\!17}{68\!\cdots\!32}a^{12}+\frac{41\!\cdots\!01}{98\!\cdots\!76}a^{11}+\frac{53\!\cdots\!17}{11\!\cdots\!72}a^{10}-\frac{88\!\cdots\!11}{68\!\cdots\!32}a^{9}-\frac{22\!\cdots\!25}{68\!\cdots\!32}a^{8}-\frac{24\!\cdots\!68}{40\!\cdots\!49}a^{7}-\frac{65\!\cdots\!19}{68\!\cdots\!32}a^{6}-\frac{67\!\cdots\!47}{68\!\cdots\!32}a^{5}+\frac{13\!\cdots\!41}{11\!\cdots\!72}a^{4}+\frac{23\!\cdots\!07}{24\!\cdots\!94}a^{3}-\frac{24\!\cdots\!81}{34\!\cdots\!16}a^{2}-\frac{44\!\cdots\!87}{28\!\cdots\!43}a+\frac{12\!\cdots\!60}{85\!\cdots\!29}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 18999317.186917692 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 18999317.186917692 \cdot 1}{6\cdot\sqrt{781553895537094671760883712}}\cr\approx \mathstrut & 1.72872571405735 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_3\times S_3^2$ (as 18T46):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.3326427.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.2900653922406629376.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(13\)
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |